Question

use an appropriate area formula to find the area of the triangle with the given side lengths.
a = 13 m
b = 8 m
c = 10 m
the area of the triangle is (square m^{2}).
(round your answer to the nearest tenth.)

Explanation:

Step1: Calculate the semi - perimeter (s)

The formula for the semi - perimeter \(s\) of a triangle with side lengths \(a\), \(b\), and \(c\) is \(s=\frac{a + b + c}{2}\).
Substitute \(a = 13\), \(b = 8\), and \(c = 10\) into the formula:
\(s=\frac{13 + 8+10}{2}=\frac{31}{2}=15.5\)

Step2: Apply Heron's formula to find the area (A)

Heron's formula for the area of a triangle is \(A=\sqrt{s(s - a)(s - b)(s - c)}\).
Substitute \(s = 15.5\), \(a = 13\), \(b = 8\), and \(c = 10\) into the formula:
First, calculate \(s - a=15.5 - 13 = 2.5\), \(s - b=15.5 - 8 = 7.5\), \(s - c=15.5 - 10 = 5.5\)
Then, \(A=\sqrt{15.5\times2.5\times7.5\times5.5}\)
Calculate the product inside the square root: \(15.5\times2.5 = 38.75\), \(7.5\times5.5 = 41.25\), and then \(38.75\times41.25=38.75\times(40 + 1.25)=38.75\times40+38.75\times1.25 = 1550+48.4375 = 1598.4375\)
Now, take the square root: \(A=\sqrt{1598.4375}\approx39.98\approx40.0\)

Answer:

\(40.0\)